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Properties

Label 2-34e2-17.9-c1-0-19
Degree 22
Conductor 11561156
Sign −0.275+0.961i-0.275 + 0.961i
Analytic cond. 9.230709.23070
Root an. cond. 3.038203.03820
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.676 + 0.280i)3-s + (1.32 − 3.20i)5-s + (0.280 + 0.676i)7-s + (−1.74 − 1.74i)9-s + (−4.37 + 1.81i)11-s − 1.46i·13-s + (1.79 − 1.79i)15-s + (3.86 − 3.86i)19-s + 0.535i·21-s + (4.37 − 1.81i)23-s + (−4.94 − 4.94i)25-s + (−1.53 − 3.69i)27-s + (1.32 − 3.20i)29-s + (−5.72 − 2.37i)31-s − 3.46·33-s + ⋯
L(s)  = 1  + (0.390 + 0.161i)3-s + (0.592 − 1.43i)5-s + (0.105 + 0.255i)7-s + (−0.580 − 0.580i)9-s + (−1.31 + 0.546i)11-s − 0.406i·13-s + (0.462 − 0.462i)15-s + (0.886 − 0.886i)19-s + 0.116i·21-s + (0.911 − 0.377i)23-s + (−0.989 − 0.989i)25-s + (−0.294 − 0.711i)27-s + (0.246 − 0.594i)29-s + (−1.02 − 0.425i)31-s − 0.603·33-s + ⋯

Functional equation

Λ(s)=(1156s/2 ΓC(s) L(s)=((−0.275+0.961i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1156s/2 ΓC(s+1/2) L(s)=((−0.275+0.961i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11561156    =    22⋅1722^{2} \cdot 17^{2}
Sign: −0.275+0.961i-0.275 + 0.961i
Analytic conductor: 9.230709.23070
Root analytic conductor: 3.038203.03820
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1156(757,⋅)\chi_{1156} (757, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1156, ( :1/2), −0.275+0.961i)(2,\ 1156,\ (\ :1/2),\ -0.275 + 0.961i)

Particular Values

L(1)L(1) ≈\approx 1.5520064631.552006463
L(12)L(\frac12) ≈\approx 1.5520064631.552006463
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
17 1 1
good3 1+(−0.676−0.280i)T+(2.12+2.12i)T2 1 + (-0.676 - 0.280i)T + (2.12 + 2.12i)T^{2}
5 1+(−1.32+3.20i)T+(−3.53−3.53i)T2 1 + (-1.32 + 3.20i)T + (-3.53 - 3.53i)T^{2}
7 1+(−0.280−0.676i)T+(−4.94+4.94i)T2 1 + (-0.280 - 0.676i)T + (-4.94 + 4.94i)T^{2}
11 1+(4.37−1.81i)T+(7.77−7.77i)T2 1 + (4.37 - 1.81i)T + (7.77 - 7.77i)T^{2}
13 1+1.46iT−13T2 1 + 1.46iT - 13T^{2}
19 1+(−3.86+3.86i)T−19iT2 1 + (-3.86 + 3.86i)T - 19iT^{2}
23 1+(−4.37+1.81i)T+(16.2−16.2i)T2 1 + (-4.37 + 1.81i)T + (16.2 - 16.2i)T^{2}
29 1+(−1.32+3.20i)T+(−20.5−20.5i)T2 1 + (-1.32 + 3.20i)T + (-20.5 - 20.5i)T^{2}
31 1+(5.72+2.37i)T+(21.9+21.9i)T2 1 + (5.72 + 2.37i)T + (21.9 + 21.9i)T^{2}
37 1+(10.5+4.38i)T+(26.1+26.1i)T2 1 + (10.5 + 4.38i)T + (26.1 + 26.1i)T^{2}
41 1+(2.29+5.54i)T+(−28.9+28.9i)T2 1 + (2.29 + 5.54i)T + (-28.9 + 28.9i)T^{2}
43 1+(−8.76−8.76i)T+43iT2 1 + (-8.76 - 8.76i)T + 43iT^{2}
47 1−6.92iT−47T2 1 - 6.92iT - 47T^{2}
53 1+(0.656−0.656i)T−53iT2 1 + (0.656 - 0.656i)T - 53iT^{2}
59 1+(6.69+6.69i)T+59iT2 1 + (6.69 + 6.69i)T + 59iT^{2}
61 1+(−2.85−6.89i)T+(−43.1+43.1i)T2 1 + (-2.85 - 6.89i)T + (-43.1 + 43.1i)T^{2}
67 1+1.07T+67T2 1 + 1.07T + 67T^{2}
71 1+(2.02+0.840i)T+(50.2+50.2i)T2 1 + (2.02 + 0.840i)T + (50.2 + 50.2i)T^{2}
73 1+(−0.765+1.84i)T+(−51.6−51.6i)T2 1 + (-0.765 + 1.84i)T + (-51.6 - 51.6i)T^{2}
79 1+(1.66−0.690i)T+(55.8−55.8i)T2 1 + (1.66 - 0.690i)T + (55.8 - 55.8i)T^{2}
83 1+(−6.69+6.69i)T−83iT2 1 + (-6.69 + 6.69i)T - 83iT^{2}
89 1+9.46iT−89T2 1 + 9.46iT - 89T^{2}
97 1+(3.41−8.24i)T+(−68.5−68.5i)T2 1 + (3.41 - 8.24i)T + (-68.5 - 68.5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.214599605353705209776549672573, −9.008425166806110335184597092153, −8.078646350298396871417752106839, −7.25290325264930211675683651513, −5.84131242469744154993064627964, −5.27877785187410774627692358717, −4.55054788421398675815075590861, −3.14064074045583148888326586653, −2.14947643926354139004776962426, −0.60935028652092943295597539169, 1.85338561561460647775937631935, 2.89165020422722743005005287679, 3.43580425308207998314122047496, 5.19518847943290540026143914810, 5.72260923823222201682328123736, 6.93273807060848877981025925003, 7.44366683984468689570205352387, 8.315754181870875271774040354551, 9.213473178678705938078345683067, 10.32220166253127270955223204113

Graph of the ZZ-function along the critical line